Determine the gradient and y-intercept for the given equations of straight lines.

$$y = \frac{6x}{2} + \frac{4}{2}$$ $$y = 3x + 2$$

The gradient $m$ is the coefficient of $x$.
The gradient is $\boxed{3}$.

b) $x + 2y = 2$

$$y = -\frac{1}{2}x + \frac{2}{2}$$ $$y = -\frac{1}{2}x + 1$$

The gradient $m$ is the coefficient of $x$.
The gradient is $\boxed{-\frac{1}{2}}$.

c) $2y = 4x$

$$y = \frac{4x}{2}$$ $$y = 2x$$

The gradient $m$ is the coefficient of $x$.
The gradient is $\boxed{2}$.

Question 3: Determine the y-intercept in each of the following equations of straight lines:

a) $y = 3x - 6$

The equation is already in the form $y = mx + c$, where $c$ is the y-intercept.
The y-intercept is $\boxed{-6}$.

b) $x + 3y - 4 = 0$

Subtract $x$ and add 4 to both sides.

$$3y = -x + 4$$

$$y = -\frac{1}{3}x + \frac{4}{3}$$

The y-intercept $c$ is the constant term.
The y-intercept is $\boxed{\frac{4}{3}}$.

c) $\frac{1}{2}y = x - 3$

$$y = 2(x - 3)$$ $$y = 2x - 6$$

The y-intercept $c$ is the constant term.
The y-intercept is $\boxed{-6}$.

d) $y = \frac{1}{2}x + 5$

The equation is already in the form $y = mx + c$, where $c$ is the y-intercept.
The y-intercept is $\boxed{5}$.

Question 4: Determine the gradient of a line with an equation $y = 4x + 5$.

The equation is already in the form $y = mx + c$, where $m$ is the gradient.
The gradient is $\boxed{4}$.

Question 5: The equation of a line is given as $2x + y = 8$. Determine:

a) the gradient of the line,

Subtract $2x$ from both sides.

$$y = -2x + 8$$

The gradient $m$ is the coefficient of $x$.
The gradient is $\boxed{-2}$.

b) the value of the y-intercept.

From the equation $y = -2x + 8$, the y-intercept $c$ is the constant term.
The y-intercept is $\boxed{8}$.

Question 6: Given a line with an equation $\frac{2}{3}x + \frac{5}{7}y = 1$.

a) Express the equation in the form $y = mx + c$.

$$y = \frac{7}{5} \left( 1 - \frac{2}{3}x \right)$$ $$y = \frac{7}{5} - \frac{14}{15}x$$ $$y = -\frac{14}{15}x + \frac{7}{5}$$

The equation in the form $y = mx + c$ is $\boxed{y = -\frac{14}{15}x + \frac{7}{5}}$.

b) Determine the gradient of the line.

From the equation $y = -\frac{14}{15}x + \frac{7}{5}$, the gradient $m$ is the coefficient of $x$.
The gradient is $\boxed{-\frac{14}{15}}$.

c) The value of y-intercept.

From the equation $y = -\frac{14}{15}x + \frac{7}{5}$, the y-intercept $c$ is the constant term.
The y-intercept is $\boxed{\frac{7}{5}}$.